## Abstract

We consider the focusing performance of a radially polarized sinh-Gaussian beam. The sinh-Gaussian beam can be considered as superposition of a series of eccentric Gaussian beam. Based on the Richards-Wolf formulas, high beam quality and subwavelength focusing are achieved for the radially polarized incident sinh-Gaussian beam. Therefore, sinh-Gaussian beam can be applied in the focusing system with high numerical aperture to achieve focusing with superresolution.

© 2013 Optical Society of America

## 1. Introduction

With the development of modern microscopical and micro-fabrication technique, optical imaging system and optical tomography generally call for high lateral resolution [1–4]. However, when propagation of light wave is perturbed, such as encountering an obstacle or focusing by phase element, diffraction occurs. Based on Abbe’s principle, airy disk is formed at the focal plane of lens [5]. For an aplanatic imaging lens with numerical aperture NA, the finite focusing spot size of plane incident light beam is governed by 0.51*λ*/NA, where *λ* is the wavelength of incident light. However, in order to obtain small spot size, the wavelength *λ* can not be shortened unlimitedly and the effective NA is limited by lens structure even for immersion lens system. Therefore, in the case of special wavelength *λ* and NA of optical imaging system, achieving a higher lateral resolution of focusing spot of an objective or focusing optical system is still challenging in the field of optical direct writing, optical data storage and so on [6–8].

In order to obtain focusing spot with higher resolution, several methods have been proposed based on far field apodization technique and handling of evanescent waves with near field diffraction structure [5, 6, 9, 10]. For example, binary phase filter has been widely studied and the structured illumination has been applied to optical lithography and confocal microscopy [5]. However, as the key property of optical beam, the polarization state of incident light should be considered [6,9–13]. Therefore, based on polarization of light, a variety of filter were proposed to improve focusing characteristics [1, 11, 14]. For example, Dorn *et al* firstly experimentally demonstrated that a radially polarized incident beam can be focused to a spot size significantly smaller than for linear polarization [11]. For a high NA objective, the needle of longitudinally polarized light was achieved by using binary optics for the first time [6]. For radially polarized beam, the focus of research is the design of the binary phase or complex amplitude apodizers [6]. However, the mode or intensity profile of incident beam only be considered to achieve focusing with high resolution recently. For example, the sharper focal spot was achieved by higher order radially polarized Laguerre-Gaussian beam and hollow Bessel-Gaussian beam [15–18].

Recently, one kind of Hermite-sinusoidal-Gaussian beams, which named sinh-Gaussian beam, has been received intensive attentions [19]. The similarity of hollow sinh-Gaussian (HsG) beam and higher-order Laguerre-Gaussian beam is the intensity null at center. Therefore, a radially polarized HsG beam will show similar focusing characteristics comparing with higher-order Laguerre-Gaussian beam. In this paper, we demonstrate the focusing performance of radially polarized hollow sinh-Gaussian beam by high NA objective lens based on Richards and Wolf’s theory [20]. Our calculated results indicate that subwavelength focusing can be obtained and one can achieve longitudinal polarized focusing simultaneously.

In Sec. 2, the hollow sinh-Gaussian beam and Richards-Wolf formulas are introduced. The numerical results of the focusing radially polarized sinh-Gaussian beam are analyzed in Sec. 3. Some discussions and conclusion are given in Sec. 4.

## 2. Hollow sinh-Gaussian beams and Richards-Wolf formulas

For a high NA objective lens, the electric field of hollow sinh-Gaussian incident beam at a pupil can be defined as follows [19]:

*m*(

*m*= 0, 1, 2, ···) is the order of the hollow sinh-Gaussian beam. Obviously, for

*m*= 0, the beam governed by Eq. (1) is conventional Gaussian beam. However, a new kind of sinh-Gaussian beam is obtained when

*m*is greater than 1.0.

*θ*is determined by NA of objective lens and 0 ≤

*θ*≤ arcsin(NA/

*n*). Where,

*n*= 1.0 is the index of refraction of free space. Based on series, Eq. (1) can be considered as the sum of hollow Gaussian beam. Number of hollow Gaussian beam and its characteristic are determined by parameter

*m*and

*ω*

_{0}, simultaneously.

Obviously, as defined in Eq. (1), the amplitude distribution of hollow sinh-Gaussian beam is determined by *ω*_{0} and *m*. In order to describe the relation of the amplitude distribution and the parameters of sinh-Gaussian intuitively, the normalized amplitude distribution of sinh-Gaussian with different value of parameter *ω*_{0} and *m* are shown in Fig. 1 for focusing lens with NA = 0.95. It is easy to see that the position of maximal amplitude of hollow sinh-Gaussian beam is shifted to right while the value of *ω*_{0} or *m* are increasing. Therefore, one can control the amplitude distribution of hollow sinh-Gaussian beam by choosing *ω*_{0} or *m* reasonably.

For radially polarized sinh-Gaussian incident beam, the electric field near the focus *z* = 0 of high NA objective lens can be calculated by using the Richards-Wolf formulas [20]. For radial polarized sinh-Gaussian beam governed by Eq. (1), the focusing field is simplified as [6]:

*E*(

_{r}*r*,

*z*) and

*E*(

_{z}*r*,

*z*) are radial and longitudinal electric field component near the focus

*z*= 0, respectively. Considering that

*E*(

_{r}*r*,

*z*) and

*E*(

_{z}*r*,

*z*) are orthogonal, the total electric energy density is given as ${E}_{\text{t}}^{2}\left(r,z\right)={E}_{r}^{2}\left(r,z\right)+{E}_{z}^{2}\left(r,z\right)$. In order to describe the focusing property, the full-width at half-maximum (FWHM) at focal plane will be investigated. Meanwhile, in order to describe longitudinal component of the focused field, the beam quality

*η*is defined as

*η*= Φ

*/(Φ*

_{z}*+ Φ*

_{z}*), where ${\mathrm{\Phi}}_{i}=2\pi {\int}_{0}^{{r}_{0}}{\left|{E}_{i}\left(r,0\right)\right|}^{2}r\text{d}r$ (*

_{r}*i*=

*r*and

*z*) and

*r*

_{0}is the first zero point in the distribution of focusing electric density at focal plane, respectively. However, in order to include total energy of incident beam, the upper range of integration is taken as 4

*λ*, which is larger than

*r*

_{0}, for the calculation of

*η*in this paper.

## 3. Focusing property of a radially polarized sinh-Gaussian beam

In order to obtain the focusing property of a radially polarized sinh-Gaussian beam, the numerical aperture of the focusing lens is considered as NA = 0.95 and *n* = 1 is the index of refraction of free space. Therefore, *α* is approximately equal to 71.8° for Eqs. (2) and (3). Based on Eqs. (2) and (3), the focusing performance of the radially polarized sinh-Gaussian beam is shown in Fig. 2. Figures 2(a1), 2(b1) and 2(c1) display the lateral normalized intensity distribution on focal plane *z* = 0. From Figs. 2(a1), 2(b1) and 2(c1), one can see that focusing is achieved in Figs. 2(b1) and 2(c1). In the case of (0.125, 4), lens with NA = 0.95 fail to focus the incident radially polarized sinh-Gaussian beam at the focal spot. As shown in Figs. 2(a2)–2(a4), 2(b2)–2(b4), and 2(c2)–2(c4), one can easily know that the total focused field is determined by radial and longitudinal components. The lateral intensity distribution of radial polarized component has a intensity null on the axial or on focal plane. Moreover, the position of maximum of radial component is most near the axis. However, the longitudinal component has a intensity maximum on the axis. Therefore, the maximal intensity of radial and longitudinal component directly determine the distribution and spot size of focused field. If the maximum of intensity occurs in radial component, one can not obtain focusing at focal spot, which shown as black curve in Fig. 2(a1) or contour plot of Fig. 2(a4). Otherwise, one can achieve beam focusing at focal plane. The amplitude distribution of cross section of incident sinh-Gaussian beams used in Fig. 2 are shown in Fig. 1. The black, red curve in Fig. 1(b), and the pink curve in Fig. 1(a) are governed by (0.125, 4), (0.25, 4) and (0.25, 8), respectively. Obviously, the radius of incident sinh-Gaussian beam is increasing gradually from Figs. 2(a1), to 2(b1), and to 2(c1). In Figs. 2(a1)–(a3), the incident beam is described as black curve in Fig.1(b). Obviously, the amplitude of sinh-Gaussian beam is approximately zero, when *θ* is larger than 0.6 rad. Therefore, it can be considered as focusing by lens with low NA. In the view of transfer function, the incident beam described by black curve in Fig. 1(b) include much of components with low frequency. Therefore, the focusing field exhibits large lateral width at focal plane. The high frequency components of incident beam is increasing for large *ω*_{0} and *m* in Eq. (1). Therefore, the circular focusing spot appears and the lateral size of the focusing spot is decreasing for large *ω*_{0} and *m*.

According to Eq. (2), the intensity of radial component at focal spot is zero while *J*_{1}(0) is zero. It is shown by red solid curves in Figs. 2(a1), 2(b1) and 2(c1). The maximal intensity don’t locate at focal spot will enlarge the size of focused spot. Meanwhile, in Eq. (3), *E _{z}*(0, 0) is nozero. Therefore, the focusing is determined by the weight of longitudinal component. Obviously, numerical results indicate that the weight of longitudinal electric field component at focused spot will increase for large value of parameters (

*m*,

*ω*

_{0}). Usually, longitudinal electric field component dominates the focusing field at focal plane, while

*m*> 4 and

*ω*

_{0}> 0.25 are satisfied, simultaneously. Actually, for fixed value of

*m*(or

*ω*

_{0}), the focused field and subwavelength focused spot can be achieved if large value of

*ω*

_{0}(or

*m*) is selected. However, extreme large value of

*m*and

*ω*

_{0}is not recommended based on the numerical results. It is worth noting that incident energy should be confined in the pupil of lens when one determined the value of parameter of sinh-Gaussian beam. Therefore, considering the previous designed annual filter, most incident energy of sinh-Gaussian beam lie in the region near the edge of the lens.

Based on results shown in Fig. 2, one can find that the resolution of focusing spot is increasing with the increasing values of parameters *ω*_{0} and *m* of incident sinh-Gaussian beam. In order to further elaborate the focusing performance of sinh-Gaussian beam focused by the lens with large NA, the focusing characteristics, such as FWHM and beam quality *η*, are tabulated in Table 1 for different value of (*ω*_{0}, *m*). In the case of (0.125, 4) and (0.250, 2), the maximal amplitude of incident sinh-Gaussian beam is near the origin of coordinate and the focusing spot is vanished at focal plane *z* = 0. Therefore, in this case, the FWHM of focusing field can’t be defined. It is obvious that the focusing field with null central intensity appears for small value of (*ω*_{0}, *m*). On the other hand, for fixed value of *ω*_{0} (or *m*), the beam quality *η* is increasing while *m* (or *ω*_{0}) is increasing. Therefore, in order to obtain longitudinal polarized focusing, larger *ω*_{0} or *m* is essential in experiment. The focusing characteristics of incident beam with large *ω*_{0} and *m* is similar with that of beam with annulus apodizer [21]. For example, in the case of (0.25, 8), focusing spot appears at focal spot with FWHM of 0.47*λ* and beam quality *η* of 83.4%. Moreover, superresolution is achieved in the case. Obviously, the FWHM and beam quality are superior to that in the case of (0.25, 6). It is more important that radial polarized sinh-Gaussian can be adopted to realize focusing with superresolution.

In the aforementioned content, the focusing performance of radial polarized sinh-Gaussian beam is discussed. Moreover, in previous researches, different kinds of incident beams, such as Bessel-Gaussian beam [6], Gaussian beam and Laguerre-Gaussian beam [17], are investigated. Therefore, comparison of focusing performance between sinh-Gaussian and other different incident beam are important in the field. Simply, as comparison with sinh-Gaussian beam, the intensity distribution at focal plane of radial polarized Bessel, Bessel-Gaussian and Gaussian beam are shown in Figs. 3(a)–(c), respectively. From Figs. 2(c1) and 3(a)–(c), one can see that radial component is compressed for sinh-Gaussian beam with (0.25, 8). The FWHM are 0.59*λ*, 0.68*λ* and 0.78*λ* for Bessel, Bessel-Gaussian [6] and Gaussian beam, respectively. However, the FWHM is 0.47*λ* in Fig. 2(c1). Although the FWHM is 1.13*λ* for sin-Gaussian beam with (0.25, 4), small FWHM can be achieved as shown in Tab. 1. Therefore, the radial sinh-Gaussian beam show the superiority in focusing with superresolution.

## 4. Conclusion

In summary, we have clarified the focusing performance of radial polarized sinh-Gaussian beam by lens with large NA based on the Richards-Wolf’s formulas. The FWHM and beam quality are analyzed for different parameters of sinh-Gaussian beam. High beam quality and subwavelength focusing are achieved, simultaneously. For small *m* and *ω*_{0}, their focusing characteristics are similar with that in focusing system with small NA. However, for large *m* and *ω*_{0}, the focusing performance is similar with that of beam with annulus apodizer. One can obtain sub-wavelength focusing spot and overcome diffraction limit. The profile of sinh-Gaussian beam is similar with hollow Gaussian beam for large *m* and *ω*_{0}. Therefore, focusing performance of sinh-Gaussian beam can be considered as that of annular pattern of incident beam and the sinh-Gaussian beam can be introduced to achieve super resolution.

## Acknowledgments

This work was funded by the Program for New Century Excellent Talents in University (Grant No. NECT100059), the National Natural Science Foundation of China (Grants No. 61008039 and 51275111), the Research Fund for the Doctoral Program of Higher Education of China (Grants No. 20102304110006 and 20102302120008) and the Fundamental Research Funds for the Central Universities (Grant No. HIT.NSRIF.2012017).

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